Our objective function \(F_p (\mathbf {U})\)  (REF )
is nonconvex for \(p\in (1,2)\) , and thus convergence to a global minimum cannot be guaranteed. Minimizing \(F_p\)  directly for a small value of \(p\)  results, in most cases, in convergence to a nonoptimal local minimum. Therefore, we take advantage of the fact that our minimization problem (REF ) exhibits a convex behavior for \(p = 2\) , and thus the global minimizer can be computed. The fact that \(F_p\)  is also continuous in \(p\)  suggests that for close values of \(p_1, p_2\) , the solution of \(F_{p_1} (\mathbf {U}), F_{p_2} (\mathbf {U})\)  will be close as well [1]}.
Accordingly, to find a solution at a given \(p\in (1,2)\) 
we solve (REF ) by gradually reducing the value of \(p\)  (starting from \(p=2\) ),
with the solution at the current \(p\)  serving as the
initial iterate for the next \(p\) -level.
In each step, we minimize our objective with a Grassmannian Newton's method, as it has proven to have a superlinear convergence rate close to the local optima and quadratic elsewhere [2]}. The linear substeps within the Newton method are handled by a Grassmannian truncated conjugate gradient scheme.
The value of \(p\)  is decreased from two towards one. We follow the approach suggested in [1]}, with \(p_{t+1} = p_t \cdot \alpha \) , and \(\alpha = 0.9\) . The convergence of our algorithm is monitored by a combination of discrete and continuous criteria. We terminate the minimization procedure at each \(p\) -level, and move to the next one, if the differences in our objective are marginal, i.e., if \(\Vert f_p\left( \mathbf {U}_{t+1}\right) - f_p\left( \mathbf {U}_{t} \right) \Vert  \le 10^{-8}\)  for at least five Riemannian Newton iterations. Additionally, we advance to the next \(p\) -level if the differences in the gradient are negligible, i.e., if \(\Vert  \nabla f_p\left( \mathbf {U}_{t+1}\right) - \nabla f_p\left( \mathbf {U}_{t} \right) \Vert  \le 10^{-6}\)  after two iterations. Finally, in order to capture the discrete nature of our problem, we consider the process in the current \(p\) -level to be terminated if after five Riemannian Newton iterations the value of the resulting ratio cut (REF ) is left unchanged.
